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The distribution of non-Parry Perron numbers and their conjugates

  • Hachem Hichri,
  • Chris Smyth

摘要

We prove that the set of Perron numbers that are not Parry numbers is dense in the interval \((1,\infty )\) ( 1 , ) and that the set of all Galois conjugates of such numbers is dense in the whole complex plane. While the Parry numbers themselves are known also to be dense in \((1,\infty )\) ( 1 , ) , their conjugates are known from work of Solomyak and others to be much more restricted: they are confined to a subset of the disc \(|z|<(1+\sqrt{5})/2\) | z | < ( 1 + 5 ) / 2 . Our work is in response to a remark of Akiyama, drawing attention to our lack of knowledge about non-Parry Perron numbers and their conjugates.